New approaches to epidemic modeling on networks

In this article, we develop two independent and new approaches to model epidemic spread in a network. Contrary to the most studied models, those developed here allow for contacts with different probabilities of transmitting the disease (transmissibilities). We then examine each of these models using some mean field type approximations. The first model looks at the late-stage effects of an epidemic outbreak and allows for the computation of the probability that a given vertex was infected. This computation is based on a mean field approximation and only depends on the number of contacts and their transmissibilities. This approach shares many similarities with percolation models in networks. The second model we develop is a dynamic model which we analyze using a mean field approximation which highly reduces the dimensionality of the system. In particular, the original system which individually analyses each vertex of the network is reduced to one with as many equations as different transmissibilities. Perhaps the greatest contribution of this article is the observation that, in both these models, the existence and size of an epidemic outbreak are linked to the properties of a matrix which we call the R-matrix. This is a generalization of the basic reproduction number which more precisely characterizes the main routes of infection.

Automated summary

In this article, two independent and novel approaches are developed to model epidemic spread in a network. These models allow for contacts with different probabilities of transmitting the disease, contrary to most existing models. Mean field type approximations are used to examine each model. The first model calculates the probability of a given vertex being infected in the late-stage effects of an epidemic outbreak, based on a mean field approximation and the number of contacts and their transmissibilities. The second model is a dynamic model analyzed using a mean field approximation, reducing the dimensionality of the system.